10.2 Lie algebras and exponential map

Lie algebras and the exponential map are key concepts in algebraic groups. They provide a powerful tool for studying the local structure of Lie groups, connecting abstract algebra with geometry and analysis.

The exponential map bridges Lie algebras and Lie groups, allowing us to construct one-parameter subgroups and explore group properties. It's essential for understanding the structure and classification of algebraic groups, including simple and semisimple Lie algebras.

Lie algebras and algebraic groups

Definition and properties of Lie algebras

• A Lie algebra is a vector space $𝔤$ over a field $𝔽$ equipped with a binary operation called the Lie bracket, denoted $[⋅,⋅]$, satisfying:
  • Bilinearity: $[aX+bY,Z] = a[X,Z] + b[Y,Z]$ and $[X,aY+bZ] = a[X,Y] + b[X,Z]$ for $a,b ∈ 𝔽$ and $X,Y,Z ∈ 𝔤$
  • Alternativity: $[X,X] = 0$ for all $X ∈ 𝔤$
  • Jacobi identity: $[X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0$ for all $X,Y,Z ∈ 𝔤$
• Examples of Lie algebras include:
  • The vector space of $n × n$ matrices over a field $𝔽$ with the commutator bracket $[A,B] = AB - BA$ ($𝔤𝔩_n(𝔽)$)
  • The vector space of smooth vector fields on a manifold with the Lie bracket of vector fields ($𝔛(M)$)

Relation between Lie algebras and algebraic groups

• The Lie algebra $𝔤$ of an algebraic group $𝔾$ is the tangent space at the identity element, $𝔤 = 𝑇_𝑒𝔾$, with the Lie bracket given by the commutator of left-invariant vector fields
• For each Lie group homomorphism $φ: 𝔾 → ℍ$, there is an associated Lie algebra homomorphism $dφ: 𝔤 → 𝔥$ between their corresponding Lie algebras, preserving the Lie bracket structure
• The adjoint representation of a Lie group $𝔾$ on its Lie algebra $𝔤$ is given by the differential of the conjugation map, $Ad: 𝔾 → GL(𝔤)$, where $Ad(g)(X) = d(C_g)_e(X)$ for $g ∈ 𝔾$ and $X ∈ 𝔤$, and $C_g(h) = ghg^{-1}$ is the conjugation map
• Examples of the correspondence between Lie groups and Lie algebras:
  • The Lie algebra of the general linear group $GL(n, ℂ)$ is the space of $n × n$ complex matrices $𝔤𝔩_n(ℂ)$
  • The Lie algebra of the special orthogonal group $SO(n)$ is the space of skew-symmetric $n × n$ real matrices $𝔰𝔬_n$

Exponential map for Lie groups

Definition and properties of the exponential map

• The exponential map $exp: 𝔤 → 𝔾$ is a local diffeomorphism from a neighborhood of $0$ in the Lie algebra $𝔤$ to a neighborhood of the identity element in the Lie group $𝔾$
• For a matrix Lie group $𝔾 ⊆ GL(n, ℂ)$, the exponential map is given by the matrix exponential: $exp(X) = ∑_{k=0}^∞ \frac{X^k}{k!}$ for $X ∈ 𝔤$
• The exponential map satisfies the property $exp(sX)exp(tX) = exp((s+t)X)$ for $s, t ∈ ℝ$ and $X ∈ 𝔤$, making it a group homomorphism from $(𝔤, +)$ to $(𝔾, ⋅)$
• The derivative of the exponential map at $0 ∈ 𝔤$ is the identity map, $d(exp)_0 = id_𝔤$
• Examples of exponential maps:
  • For the Lie group $GL(n, ℂ)$, the exponential map is the matrix exponential $exp(A) = ∑_{k=0}^∞ \frac{A^k}{k!}$ for $A ∈ 𝔤𝔩_n(ℂ)$
  • For the Lie group $SO(3)$, the exponential map is given by the Rodrigues' rotation formula $exp(θ\hat{n}) = I + \sin θ[\hat{n}]_× + (1-\cos θ)[\hat{n}]_×^2$, where $θ\hat{n} ∈ 𝔰𝔬_3$ is a skew-symmetric matrix representing a rotation by angle $θ$ about the axis $\hat{n}$

Construction of one-parameter subgroups

• The exponential map is used to construct one-parameter subgroups of a Lie group, which are smooth homomorphisms from $(ℝ, +)$ to $(𝔾, ⋅)$ of the form $t ↦ exp(tX)$ for some $X ∈ 𝔤$
• One-parameter subgroups are the integral curves of left-invariant vector fields on the Lie group $𝔾$
• The exponential map establishes a bijection between the Lie algebra $𝔤$ and the set of one-parameter subgroups of $𝔾$
• Examples of one-parameter subgroups:
  • In the Lie group $GL(n, ℂ)$, the one-parameter subgroup generated by a matrix $A ∈ 𝔤𝔩_n(ℂ)$ is given by $t ↦ exp(tA) = ∑_{k=0}^∞ \frac{t^k A^k}{k!}$
  • In the Lie group $SO(3)$, the one-parameter subgroup generated by a skew-symmetric matrix $θ\hat{n} ∈ 𝔰𝔬_3$ represents rotations about the axis $\hat{n}$ by angles proportional to $t$

Properties and applications of the exponential map

Surjectivity and generating properties

• The exponential map is surjective for connected, simply connected Lie groups, meaning every element of $𝔾$ can be written as $exp(X)$ for some $X ∈ 𝔤$
• For compact Lie groups, the exponential map is not necessarily surjective, but its image generates a dense subset of the group
• Examples:
  • The exponential map of the Lie group $SU(2)$, the group of $2 × 2$ unitary matrices with determinant 1, is surjective because $SU(2)$ is simply connected
  • The exponential map of the Lie group $SO(3)$, the group of $3 × 3$ orthogonal matrices with determinant 1, is not surjective, but its image generates a dense subset of $SO(3)$

Baker-Campbell-Hausdorff formula

• The Baker-Campbell-Hausdorff formula expresses the product of two exponentials in a Lie group in terms of a single exponential: $exp(X)exp(Y) = exp(Z)$, where $Z$ is a series in terms of nested Lie brackets of $X$ and $Y$
• The first few terms of the Baker-Campbell-Hausdorff formula are: $Z = X + Y + \frac{1}{2}[X,Y] + \frac{1}{12}[X,[X,Y]] - \frac{1}{12}[Y,[X,Y]] + ...$
• The Baker-Campbell-Hausdorff formula is useful for approximating products in Lie groups and studying the local structure of Lie groups near the identity
• Example: In the Lie group $GL(n, ℂ)$, the Baker-Campbell-Hausdorff formula gives an approximation of the product of two matrices close to the identity: $exp(A)exp(B) ≈ exp(A+B+\frac{1}{2}[A,B])$ for small matrices $A,B ∈ 𝔤𝔩_n(ℂ)$

Structure and classification of Lie algebras

Simple and semisimple Lie algebras

• A Lie algebra is simple if it has no non-trivial ideals and is not abelian. The classification of simple Lie algebras over $ℂ$ leads to the four infinite families $A_n, B_n, C_n, D_n$, and five exceptional Lie algebras $G_2, F_4, E_6, E_7, E_8$
• A Lie algebra is semisimple if it is a direct sum of simple Lie algebras. The Killing form, a symmetric bilinear form on $𝔤$, is non-degenerate if and only if $𝔤$ is semisimple
• Examples of simple and semisimple Lie algebras:
  • The Lie algebra $𝔰𝔩_n(ℂ)$ of traceless $n × n$ complex matrices is simple for $n ≥ 2$
  • The Lie algebra $𝔰𝔬_n(ℂ)$ of skew-symmetric $n × n$ complex matrices is simple for $n ≥ 5$ and semisimple for $n = 3, 4$

Levi decomposition and solvable radical

• The Levi decomposition states that any finite-dimensional Lie algebra $𝔤$ can be written as a semidirect product $𝔤 = 𝔰 ⋉ 𝔯$, where $𝔰$ is a semisimple subalgebra (the Levi factor) and $𝔯$ is the solvable radical
• The solvable radical $𝔯$ is the maximal solvable ideal of $𝔤$, and it consists of the nilpotent and abelian parts of the Lie algebra
• The Levi decomposition is unique up to conjugation by elements of the group $exp(𝔯)$
• Example: The Lie algebra of the Poincaré group, the semidirect product of translations and Lorentz transformations in special relativity, has a Levi decomposition $𝔭 = 𝔰𝔬(1,3) ⋉ ℝ^4$, where $𝔰𝔬(1,3)$ is the Lorentz Lie algebra and $ℝ^4$ is the abelian Lie algebra of translations

Root systems and classification

• The structure of a semisimple Lie algebra is determined by its root system, a configuration of vectors in a Euclidean space satisfying certain integrality and reflection conditions
• A root system is a finite set Φ of non-zero vectors in a Euclidean space 𝔢 satisfying:
  • If α ∈ Φ, then the only multiples of α in Φ are ±α
  • For each α ∈ Φ, the reflection σ_α(β) = β - 2\frac{(β,α)}{(α,α)}α leaves Φ invariant
  • For all α, β ∈ Φ, the number 2\frac{(β,α)}{(α,α)} is an integer
• The classification of root systems leads to the classification of semisimple Lie algebras, with each simple Lie algebra corresponding to an irreducible root system
• Examples of root systems and their corresponding Lie algebras:
  • The root system $A_n$ corresponds to the special linear Lie algebra $𝔰𝔩_{n+1}(ℂ)$
  • The root system $C_n$ corresponds to the symplectic Lie algebra $𝔰𝔭_{2n}(ℂ)$